DS202303: Paßmann, Robert (2023) Logical Structure of Constructive Set Theories. Doctoral thesis, University of Amsterdam.
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Abstract
The tautologies and admissible rules of a formal system may exceed those of its underlying logic. For example, Diaconescu, Goodman and Myhill showed that any set theory containing the axioms (and schemes) of extensionality, empty set, pairing, separation and choice proves the law of excluded middleeven if that set theory is based on intuitionistic logic.
The goal of this dissertation is, roughly speaking, to study situations where this is not the case: we show that many intuitionistic and constructive set theories are loyal to their underlying logic. We say that a formal system is (propositional/firstorder) tautology loyal if its (propositional/firstorder) tautologies are exactly those of its underlying logic. We call a formal system (propositional/firstorder) rule loyal if its (propositional/firstorder) admissible rules are exactly those of its underlying logic.
Using Kripke models with classical domains, we show that intuitionistic KripkePlatek set theory (IKP) is firstorder loyal (Chapter 4). Moreover, we introduce a realisability notion based on Ordinal Turing Machines that allows us to prove that IKP is propositional rule loyal, as well (Chapter 7). This notion of realisability also lends itself to realising infinitary set theories.
We introduce blended models for intuitionistic ZermeloFraenkel set theory (IZF) to show that this system is propositional tautology loyal (Chapter 5). A variation of this technique is useful for studying the admissible rules of various constructive set theories and proving that they are propositional rule loyal (Chapter 6).
Finally, we also prove that constructive ZermeloFraenkel set theory (CZF) is firstorder tautology loyal as well as propositional rule loyal (Chapter 8). To this end, we introduce a new notion of transfinite computability, the socalled Set Register Machines. We combine the resulting notion of realisability with Beth models to show that CZF is firstorder tautology loyal.
Item Type:  Thesis (Doctoral) 

Report Nr:  DS202303 
Series Name:  ILLC Dissertation (DS) Series 
Year:  2023 
Subjects:  Computation Logic 
Depositing User:  Dr Marco Vervoort 
Date Deposited:  31 Jan 2023 13:38 
Last Modified:  30 Mar 2023 13:17 
URI:  https://eprints.illc.uva.nl/id/eprint/2233 
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